Coalescence Model for Crumpled Globules Formed in
Polymer Collapse
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Bunin, Guy, and Mehran Kardar. "Coalescence Model for
Crumpled Globules Formed in Polymer Collapse." Phys. Rev.
Lett. 115, 088303 (August 2015). © 2015 American Physical
Society
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http://dx.doi.org/10.1103/PhysRevLett.115.088303
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Detailed Terms
PRL 115, 088303 (2015)
PHYSICAL REVIEW LETTERS
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21 AUGUST 2015
Coalescence Model for Crumpled Globules Formed in Polymer Collapse
Guy Bunin and Mehran Kardar
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 29 March 2015; published 20 August 2015)
The rapid collapse of a polymer, due to external forces or changes in solvent, yields a long-lived
“crumpled globule.” The conjectured fractal structure shaped by hierarchical collapse dynamics has proved
difficult to establish, even with large simulations. To unravel this puzzle, we study a coarse-grained model
of in-falling spherical blobs that coalesce upon contact. Distances between pairs of monomers are assigned
upon their initial coalescence, and do not “equilibrate” subsequently. Surprisingly, the model reproduces
quantitatively the dependence of distance on segment length, suggesting that the slow approach to scaling is
related to the wide distribution of blob sizes.
DOI: 10.1103/PhysRevLett.115.088303
PACS numbers: 82.35.-x, 68.43.Jk, 82.35.Pq
The rapid collapse of a polymer into a dense globule is a
long-standing problem [1–12]. Such a collapse may be
triggered by changes in solvent quality, causing the polymer
to reduce its solvent–exposed surface area by forming a
dense globule. A polymer may also be condensed by active
forces, as in the rearrangement of DNA by proteins in the
cell nucleus [2]. The rapid collapse does not allow sufficient
time for formation of topological entanglements, which
abound in an equilibrated compact globule [1–4,6,13]. It is
suggested [1] that during collapse segments of the polymer
initially condense to spherelike “blobs,” which coarsen
upon contact to form larger blobs. At any given time during
the process the state is then assumed to be characterized by a
single length scale [1,7–9], e.g., the typical size of the
blobs or the width of the tube connecting the blobs [see, e.g.,
Fig. 1(a) [14]]. A central assumption is that when two blobs
coalesce they remain more or less segregated within the
newly formed structure. This is due to the slow relaxation
processes within blobs, and due to topological constraints
that prevent polymer segments forming the blobs from
freely mixing—unlike a melt of independent polymer
segments with open ends [1,15–20]. The final configuration
is thus predicted to be a constant density, self-similar,
hierarchical structure, known as the “crumpled globule”
or “fractal globule” [1–6,11]. The end-to-end distance rm of
segments of length m in the resulting globule is predicted to
scale as rm ∼ m1=d in d space dimensions (throughout the
Letter d ¼ 3). This is in contrast to the equilibrium state
reached at much later times [10], where rm ∼ m1=2 for small
m, saturating at the globule size rmax ¼ N 1=d , where N is the
length of the polymer [21].
These predictions have been tested in several simulations
of polymer compaction [2,4–6,11], which generally confirm that the rapidly collapsed state is not entangled, and is
indeed different from the equilibrium globule. However,
they do not agree upon its fractal nature. In particular, the
expected scaling rm ∼ m1=d has not been conclusively
confirmed, even with the largest size simulations (recently
0031-9007=15=115(8)=088303(5)
extended to polymers of up to 250 000 monomers [6,22]).
This implies either a very large crossover scale before
fractal behavior is clearly manifested, or else that the
collapsed state is not strictly self-similar. The large
finite-size effects are at least partly explained by partial
mixing of short polymer segments, for which the topological constraints do not apply [1,15–20,22]. More generally, various protocols for constructing crumpled globules
have been suggested [2,4–6,12,22], and it is not clear if the
different procedures yield the same state. Settling these
issues seems to require even larger simulations, or new
theoretical insights.
FIG. 1 (color online). The coalescence model combines a
simulation of coalescing spherical drops (b),(d),(e), with estimates for distances between monomer pairs in the final structure.
MD simulations (a),(c) with initial conditions identical to (b),(d)
are shown. Initial conditions are SAW for (a),(b), RW for (c),(d),
and 1D for (e). All panels show parts of the full systems. Colors
show the position of monomers along the polymer, or the average
position for drops.
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PHYSICAL REVIEW LETTERS
To this end, we propose a coarse-grained model for the
crumpled globule formed by polymer collapse, focusing
here on active compression. The evolution of blobs is
modeled by drop coalescence, while distances between
monomers in the final crumpled state are assigned without
keeping track of their individual positions. This builds upon
the topological segregation of blobs, and the slowness of
subsequent internal rearrangements, and should apply at
scales beyond those at which these conditions hold. The
results highlight a different, dynamical source of slow
convergence to a self-similar state, and draw connections to
the physics of coagulation and drop coalescence. In
particular, the assumption of a single length scale during
the collapse may need to be amended, at least when the
collapse is due to active compression.
In the spirit of the blob theory [10], the basic entity in our
model is a “drop,” an abstraction of the blob. A drop is a
uniform density sphere that contains a subset of the
monomers from the original polymer, without explicit
position assignments within the drop. Initially, every
monomer is a drop. Drops move (as detailed below) and
coalesce into larger drops immediately upon collision, i.e.,
as soon as the spheres overlap in space. Drops do not break
into smaller drops. Coalescence conserves volume, drops
of volume v1 ; v2 forming a drop of volume v1 þ v2 . (With
the monomer volume set to unity, the drop volume equals
the number of monomers it contains.) The new drop is
centered at the center of mass of the coalesced drops. The
process terminates when all drops have merged to a single
sphere. Example coalescence runs are compared with
corresponding molecular dynamics (MD) runs in Fig. 1.
Our main interest is the structure of the final collapsed
state, as captured by the distances frm g. However, in the
coarse-grained drop coalescence model we do not keep
track of the internal structure of a drop. Instead we assign
distance estimates to all pairs of monomers within a drop.
Guided by the slow internal rearrangements in the blob
picture, a distance estimate is assigned when a pair of
monomers first comes together in a coalescence event, and
is not changed thereafter. Upon coalescence of drops of
volumes v1 and v2 , v1 × v2 pairs of monomers, one from
each drop, are assigned a distance estimate. This distance is
of order ðv1 þ v2 Þ1=d, the linear size of the new drop, which
can be sampled from any distribution. As the results are
highly insensitive to this choice, we simply assign all v1 v2
pairs the “distance” ðv1 þ v2 Þ1=d . The distance estimates
satisfy triangle inequalities, and importantly are ultrametric, the tree structure reflecting the hierarchy of the collapse
process [23]. At the end, when all monomers belong to a
single drop, all monomer pairs have been assigned a
distance estimate.
To fully define the model, it remains to prescribe the
drops’ motion between coalescence events, as well as the
initial conditions. Here, we focus on a simple dynamics for
active compression in which the drop velocity is
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proportional to its distance from the origin, i.e.,
dR~ α =dt ¼ −R~ α , where R~ α is the position of the drop center.
This corresponds to overdamped motion in a harmonic
potential. It can also be viewed as a uniform compression of
space, as the distance R~ αβ ¼ R~ α − R~ β between two drops
that have not yet coalesced changes as dR~ αβ =dt ¼ −R~ αβ , so
all relative distances shrink by the same factor per unit time.
Apart from coalescence, there is no additional interaction
between drops. In particular, we do not impose polymeric
bond interactions between sequential monomers.
The initial conditions can be any configuration of the
monomer positions. Here, we use random walk (RW), selfavoiding walk (SAW), and “one-dimensional” (1D) initial
conditions. For the latter, monomers are positioned along a
line as Rj ¼ jx̂ þ ηj , where x̂ is the unit vector in the x
direction, and the ηj are independent Gaussian variables in
d dimensions with hηj i ¼ 0, hη2j i ¼ c2 , and c a number of
the order of the drop radius. Open polymers are used in all
initial conditions. The SAW initial conditions are arguably
the most natural for a polymer in a good solvent, but other
initial conditions help in obtaining additional insight. In
particular, 1D initial conditions are used only in the
coalescence process, as the strong unidirectional compression will cause large internal rearrangements after blobs
have formed within the MD.
The coalescence model is compared with MD simulations where the polymer is composed of monomers with
standard polymeric bond attraction forces Fnn , and repulsion Frep between all monomer pairs, see the Supplemental
Material [24]. Together with the same external force as in
the coalescence model, the monomers evolve as dRi =dt ¼
−Ri þ Fnn þ Frep . The MD runs are terminated when the
polymer size stops decreasing. In both the coalescence and
MD simulations we do not add noise to the dynamics, as
the entire collapse process is fast (of order ln N, see below).
Noise may further increase rearrangement processes in the
MD simulations, which we try to minimize here. A longlived, metastable state is obtained, due to the time-scale
separation between the fast collapse process and further
rearrangements that are much slower. (As noted above,
other crumpled globule metastable states may also exist
[4–6,22], at later times.)
Now, let dcoal
be the distance estimate between monoij
mers i; j in the coalescence model, and dMD
ij ¼ ∥Ri − Rj ∥
the corresponding distance between the i; j monomers
in the final state of the MD run. In what follows we apply
MD
the same analysis to dcoal
ij , dij , referred jointly as dij .
We
define the
ffi normalized end-to-end distance rm ≡
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C hd2ij iji−jj¼m , where the average h iji−jj¼m is over all
monomer pairs separated along the polymer by m monomers, as well as over repeated runs of the two models, with
initial conditions resampled for each run. The normalization C is chosen so that
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PRL 115, 088303 (2015)
X
m
PHYSICAL REVIEW LETTERS
N−m
r2 ¼ 1.
NðN − 1Þ=2 m
(N − m is simply the number of pairs that are m units apart
in an open polymer of length N). In this way the MD and
coalescence models can be compared without any fitting
parameters.
Figure 2(a) compares rm from the coalescence model
with MD simulations for SAW initial conditions. MD
results are shown for different polymer lengths N.
Strikingly, the MD and coalescence models agree quantitatively over the entire range where the MD results of
different polymer sizes collapse, for 10−3 ≲ m=N ≤ 1. In
the small m regime, where MD simulations with different N
do not collapse, they show a different behavior (consistent
with a random walk) not present in the coalescence model.
This demonstrates that the coalescence process indeed
represents a coarse-grained model, capturing large-scale
behavior while incorporating different microscopic details.
The absence of polymeric bonds in the coalescence model
removes short-scale constraints that (as supported by the
MD simulations) do not effect large-scale properties
through the distance assignment.
The m1=3 trend line in Fig. 2(a) indicates that the
expected scaling is not present at the tested system sizes
N ≤ 25 × 103 . To more carefully assess self-similarity, we
study finite-size scaling in the standard form
rm ¼ ðm=NÞ1=3 fðm=NÞ;
ð1Þ
expected to hold for 1 ≪ m. (The additional N −1=3 reflects
the choice of normalization.) The scaling function fðxÞ
should go to a constant for x → 0, such that rm ∝ ðm=NÞ1=3
for 1 ≪ m ≪ N. Data for both MD and coalescence
models, for different initial conditions and system sizes,
are summarized in Fig. 2. The difference between MD and
coalescence results for the RW initial conditions is larger
FIG. 2 (color online). Comparison of the coalescence model
and MD results. (a) Normalized end-to-end distance rm , with
SAW initial conditions. Inset: rm =ðm=NÞ1=3 for the same rm .
(b) rm =ðm=NÞ1=3 for RW initial conditions. (c) rm =ðm=NÞ1=3 for
the coalescence model with different initial conditions and
system sizes N.
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than for the SAW. Importantly, for both SAW and RW
initial conditions, and for both models, the expected
condition, fðm=N → 0Þ ¼ const, is not seen clearly for
tested values of N; in all cases, the maximum of fðm=NÞ ¼
rm =ðm=NÞ1=3 appears to grow with increasing N. This
effect is almost absent for the 1D initial conditions, and
largest in the RW case. Since the coalescence model
includes only a minimal set of microscopic details, it is
surprising to see such a slow approach to the expected
scaling behavior. Understanding this trend in the coalescence model should provide insights into the more complex
case of the collapsing polymer.
While simplified, a full understanding of the coalescence
model—including the distribution of distance estimates
assigned as a function of the time t and separation m—is
still difficult. Fortunately, some insights regarding scaling
(or lack thereof) can be gained by examining the distribution of drop volumes as a function of time, even without
making reference to the distance estimate assignments. In
particular, the volume distributions already show a slow
approach to scaling.
Let ρt ðvÞ be the distribution ofR drop sizes at time t.
Volume
conservation implies that dvvρt ðvÞ ¼ N, while
R
dvρt ðvÞ is the number of drops at time t (averaged over
repeated runs). Within the scaling picture, the dynamics of
the collapse is described by a single typical drop size as a
function of time, v̄ðtÞ, when 1 ≪ v̄ðtÞ ≪ N. For example,
in the tube picture v̄1=d may be the thickness of the tube. In
such a scaling regime, ρt ðvÞ should depend on time only
through v̄ðtÞ as
N
v
ρt ðvÞ ¼
:
ð2Þ
ϕ
v̄ðtÞ
v̄ðtÞ2
R
The factor v̄−2 ensures that dvvρt ðvÞ remains constant.
P 2
We measure
P
P 2the typical size via v̄ðtÞ ≡ h n vn i=
n vn ¼ h n vn i=N, where the sums run over all drops
at time t, and the average h i is over initial conditions.
This definition, standard in the drop coalescence and
coagulation literature [28–31], addresses possible divergences of ρt ðvÞ at small volumes (see below). To test
Eq. (2), we plot ϕt ðvÞ ≡ N −1 v̄ðtÞ2 ρt ðvÞ against the normalized volume v=v̄ðtÞ, and check for data collapse at
different values of t.
The distributions, depicted in Figs. 3(a) and 3(b) for 1D
and RW initial conditions, respectively, are quite different.
The one in Fig. 3(a) (1D initial conditions) is concentrated
in the region 0.4 ≲ v=v̄ ≲ 1.6, and is strongly suppressed
outside this interval. Thus, at any given time all volumes are
of the same order, with a ratio of about 4 between the
largest and smallest. The distributions at different times
collapse nicely when plotted against v=v̄ðtÞ. In contrast, the
distributions for RW initial conditions in Fig. 3(b) are very
wide (note the log-log scale), with possibly a diverging tail
at small volumes, v=v̄ ≪ 1. (The dashed line, x−1 , is
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is perhaps most similar to our model. There, the distribution
of drop sizes is highly polydisperse when v̄ðtÞ (in our
notation) grows exponentially in time [28]. Aggregating
systems are commonly studied through the (mean-field)
Smoluchowski equation [30,31]:
Z
Z
1
0
∂ t ρðvÞ ¼
dv Jt;v0 ;v−v0 − dv0 J t;v0 ;v :
ð3Þ
2
FIG. 3 (color online). Drop volume distributions in the coalescence model. (a) 1D initial conditions. Dashed line: sequential
collisions theory. (b) RW initial conditions. Dashed line: x−1
trend line. (c) SAW initial conditions. (d) v̄ðtÞ for the three
models. Dashed lines: ewt for different cases (see text).
included as an indication of such potential divergence.)
Moreover, the distributions fail to collapse in this tail. With
SAW initial conditions [Fig. 3(c)] the results appear to be
intermediate between the above two cases, with a distribution that is finite at v ¼ 0 (at least for tested N). The
simulations in Figs. 3(a)–3(c) were carried out with
N ¼ 2.5 × 104 , 5 × 104 , and 105 , respectively, to allow
for better scaling in Figs. 3(b) and 3(c).
The evolution of v̄ðtÞ is depicted in Fig. 3(d). For 1D and
SAW initial conditions v̄ðtÞ ∝ ewt is a good fit at intermediate times, where w ¼ d=ðdν0 − 1Þ with ν0 describing
ð0Þ
the scaling rm ∼ mν0 in the initial condition. This form is
explained as follows. At time t, segments of initial length
v̄ðtÞ form drops of volume v̄ðtÞ, with diameter ½v̄ðtÞ1=d.
The initial end-to-end distance of these segments scales as
½v̄ðtÞν0 . If this reduction in length follows the general
exponential approach of free monomers [32], then
½v̄ðtÞν0 e−t ¼ ½v̄ðtÞ1=d , leading to the above relation [33].
For RW initial conditions the growth of v̄ðtÞ follows this
form for a narrower interval, with wide crossover regions,
probably related to the lack of scaling observed in Fig. 3(b).
The difference between the three volume distributions in
Fig. 3 can be directly observed in the coalescence and MD
runs in Fig. 1; note especially the large number of small
drops for RW initial conditions in Fig. 1(d). In the MD
starting from the same initial conditions, these manifest as
open segments of the polymer, alongside large condensed
regions, see Fig. 1(c); the thickness of a putative “tube” is
highly uneven. We now present theoretical approaches to
analogous problems leading to the two very different
volume distributions obtained in Figs. 3(a) and 3(b).
Broad distributions, with power-law tails at small
volumes, as in Fig. 3(b) for RW initial conditions, appear
in related problems such as diffusion-limited aggregation
[34], and drop coalescence [28,29]. Heterogeneous drop
coalescence, where drops grow but no new drops are added,
The first term is the change in density ρt ðvÞ due to
creation of drops of size v, while the second term
describes their removal due to coalescence. In the current
context, this approximation postulates a rate Jt;v1 ;v2 ¼
K v1 ;v2 ρt ðv1 Þρt ðv2 Þ for collisions between drops of volumes
v1 ; v2 , with a kernel K v1 ;v2 depending only on the volumes
of the coalescing drops. Here, rather than explicitly constructing an approximate kernel, we refer to extensively
studied scaling solutions of the Smoluchowski equation
[30,31]. In particular, for homogeneous kernels such that
K av1 ;av2 ¼ aλ K v1 ;v2 for a > 0, it is known that when
v̄ðtÞ ∝ ewt , as in our case, the scaling function ϕðxÞ in
Eq. (2) has the following properties: it is strongly (exponentially) decaying for x ≫ 1, and has a diverging powerlaw tail ϕðxÞ ∼ x−τ for 1 ≫ x, with 1 ≤ τ < 2 (the value of
τ depends on the kernel). For example, the kernel K v1 ;v2 ¼
v1 þ
2 ffi admits an exact scaling solution ϕðxÞ ¼
pvffiffiffiffiffi
ð1= 2π Þx−3=2 e−x=2 with v̄ ¼ e2t , so that τ ¼ 3=2.
Interestingly, a slow approach to the asymptotic scaling
is well documented for certain cases with such smallvolume tails [31,35–37]. The nature of this slow approach
is still not well understood, and might be sensitive to the
kernel form and initial conditions. It is intriguing to
speculate on its relation to the present problem.
In the case of 1D initial conditions in Fig. 3(a),
essentially all drops at a given time have volumes of the
order of v̄ðtÞ. Unlike RW initial conditions, all collisions
here are sequential, i.e., between drops composed of
adjacent segments along the polymer. Here, geometry
matters: after a collision gaps are formed on both sides
of the newly constructed drop, greatly reducing its probability of coalescing again before other drops have time to
grow. Smaller drops leave smaller gaps, and have an
increased probability of additional collisions. These effects
are discussed in a quantitative way in the Supplemental
Material [24]. An approximate evolution equation is
derived that admits the scaling solution shown in
Fig. 3(a) (dashed line), which strongly decays outside a
narrow interval of v=v̄, just like the results from the full
coalescence model.
The coalescence model proposed here does away with
several microscopic details present in the collapsing polymer that could delay the asymptotic approach to scaling.
The lack of simple scaling in coalescence thus points to
deeper problems with the simple model of hierarchical
collapse, in particular in the assumption of a sharply
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defined blob scale. The coalescence model is interesting in
its own right; it is closely related to widely studied
problems of coarsening of growing water drops, but differs
in the initial (polymeric) distribution of droplets. As
demonstrated, different such initial conditions lead to
widely dissimilar probability distributions. Probing the role
of fractal initial conditions in coalescence problems should
thus be worthy of further exploration.
We thank Bernard Derrida, Maxim Imakaev, Leonid
Mirny, and Adam Nahum for valuable discussions. This
research was supported by the NSF through Grant
No. DMR-12-06323. G. B. acknowledges the support of
the Pappalardo Fellowship in Physics.
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